New globular clusters may form in the merger of two galaxies. Perhaps the best examples of merger remnants are the set of ellipticals with kinematically distinct cores. Here we present Hubble Space Telescope (HST) Wide Field and Planetary Camera 2 (WFPC2) imaging of 14 kinematically distinct core ellipticals to examine their globular cluster systems. In particular, we probe the galaxy central regions, for which we might expect to see the strongest signatures of some formation and destruction processes. These data increase substantially the number of extragalactic globular cluster systems studied to date. We have developed a method for galaxy subtraction and selection of globular clusters which results in about 200 globulars per galaxy to a limiting magnitude of V similar to 25. Simulations of artificial globulars are described also. We find that the globular cluster luminosity, and color, vary only weakly, if at all, with galactocentric distance. The mean colors of globular clusters are constant with globular cluster magnitude. Several clear trends are also present. First, globular cluster colors are bluer (more metal poor by similar to 0.5 dex) than the underlying galaxy starlight at any given galactocentric distance. Second, we find a good correlation over roughly 10 magnitudes between the mean globular cluster metallicity and parent galaxy luminosity of the form Z proportional to L(0.4). This relationship includes dwarf ellipticals, spiral galaxy bulges, and giant ellipticals. Third, we find that globular cluster surface density distribution can be described by a core model, for which the core radius correlates with galaxy luminosity. Last, for the sample as a whole, the globular cluster systems are closely aligned with the galaxy major axis and are slightly rounder than the galaxy itself, although their are some notable exceptions. Our results favor scenarios in which ellipticals form from massive, gas rich progenitors at early epochs. Detailed simulations of the formation of globular cluster systems would be valuable to draw firmer conclusions.